$A$ and $B$ are infinite sets, is it true that the cardinality of its Cartesian product ($|A×B|$) is equal to max $(|A|,|B|)$?

Question

If $A$ and $B$ are infinite sets, is it true that the cardinality of its Cartesian product ($|A \times B|$) is equal to max $(|A|,|B|)$? If it is true, why? (We assume the axiom of choice)

Answer 1

Correct me please if I'm not right.

1. $|A \times B| \geq \max (|A|,|B|)$

2. $ |A \cup B| = \max (|A|, |B|)$

3. $| (A \cup B) \times (A \cup B)| = |A \cup B| = \max (|A|, |B|)$

4. $|A \times B| \leq | (A \cup B) \times (A \cup B)| = \max (|A|, |B|)$

5. $\max (|A|, |B|) \leq |A \times B| \leq \max (|A|, |B|)$

then $|A \times B| = \max (|A|, |B|)$